Five identical rectangles are arranged to form a larger rectangle $PQRS$, as shown.  The area of $PQRS$ is $4000$.  What is the length, $x$, rounded off to the nearest integer? [asy]
real x = 1; real w = 2/3;

// Draw outer square and labels
pair s = (0, 0); pair r = (2 * x, 0); pair q = (3 * w, x + w); pair p = (0, x + w);
draw(s--r--q--p--cycle);
label("$S$", s, SW); label("$R$", r, SE); label("$Q$", q, NE); label("$P$", p, NW);

// Draw other segments
draw((x, 0)--(x, w));
draw((0, w)--(2 * x, w));
draw((w, x + w)--(w, w)); draw((2 * w, x + w)--(2 * w, w));

// length labels
pair upper = (-0.1, x + w);
pair lower = (-0.1, w);
draw(lower--upper);
draw((-0.1 - 0.03, x + w)--(-0.1 + 0.03, x + w));
draw((-0.1 - 0.03, w)--(-0.1 + 0.03, w));
label("$x$", midpoint((-0.1, w)--(-0.1, x + w)), W);

pair left = (0, -0.1); pair right = (x, -0.1);
draw((0, -0.1 + 0.03)--(0, -0.1 - 0.03));
draw((x, -0.1 - 0.03)--(x, -0.1 + 0.03));
draw(left--right);
label("$x$", (x/2, -0.1), S);

[/asy]
Answer: Let $w$ be the width of each of the identical rectangles. Since $PQ=3w$, $RS=2x$ and $PQ=RS$ (because $PQRS$ is a rectangle), then $2x = 3w$, or $$w=\frac{2}{3}x.$$ Therefore, the area of each of the five identical rectangles is $$x\left(\frac{2}{3}x\right)=\frac{2}{3}x^2.$$ Since the area of $PQRS$ is 4000 and it is made up of five of these identical smaller rectangles, then $$5\left(\frac{2}{3}x^2\right)=4000$$ or $$\frac{10}{3}x^2 = 4000$$ or $x^2 = 1200$ or $x \approx 34.6$ which, when rounded off to the nearest integer, is $\boxed{35}.$